Edit: This has gotten a lot easier since they lowered her fuel consumption, as you can now essentially stay in the air forever.
Unlike most hero multi-kill achievements, Pharah's doesn't NECESSARILY have to be done with her Ultimate, but obviously it helps. Pharah is built around staying in the air and being hard to hit.
In order to maximize air time, your best bet it to start a combo with your Ult and get as many kills as possible. As this is an objective based game, you're bound to end up at a point where people will be clustered on an objective.
Once your Ult has been used (and if you're still alive), keep tapping your fuel to stay in the air as long as you can.
Once your fuel is out, use your Jump Jet to jump back into the air before hitting the ground, and you should have some fuel by the time you're back in the air. This will maximize your air time.
All the Capture maps are fairly good for this achievement since they all have at least 1 Capture Point that has a cliff Pharah can knock people into with Concussive Blast. Ilios has some cliffs at all 3 points, though the one in the ruins is only going to catch flankers. Nepal has the big indoor area. Lijiang tower has cliffs at all 3 though most people don't go outside at the major indoor part. Nothing beats that feeling of knocking multiple people off a cliff.
Escort (defense) it's going to be slightly harder since her ultimate is a bit more over time than some heroes, and if they're running around the payload enough they could heal a good bit while dodging you.
Now it's entirely possibly to get all 4 kills with Barrage but this is just to get you as many opportunities for kills as possible. Even if you are in a tight space, always try jumping before your Barrage, as it could easily turn into a Quad Kill. People don't generally expect a tight quarters Barrage.
While her Barrage does damage over a fairly wide area, it can help to be closer to the ground to make sure you focus a target down better. If you just shoot over a wide area, a Lucio could easily keep a team healed. Focus down any healers, then go for the other squishy characters. In my video though it does seem to kill a lot faster than I had anticipated from long range.
Pharah's Rockets can 2 hit most heroes if you can get a direct hit, though the splash damage if you miss is pretty terrible.
As with most Multi-Kill achievements, teaming up with a Zarya or Mei Ultimate is a great benefit.